Personally I think he was just trying to get away, he'd already been shot and severely injured by then.
He was either trying to get away from Jeremy. Or followed him as he knew he was going downstairs to reload before returning.
It's a natural reaction to run away when someone is shooting you. In the dark Neville would not know who it was, or what sort of gun it was. It could have been a shot gun, which would blow his head off, or it could have been a rifle which holds up to 11 bullets.
The other reasons to run downstairs was to arm himself, find a defensive weapon. Or to try to get outside, the Foakes's were only a few yards away.
Bob Woffinton said Neville ran downstairs to telephone the police. After Jeremy had disabled all phones at WHF. This I do not believe as there was no time to be making phone calls.